![[PDF]](/pb-assets/Icons/adobe-pdf-icon-1498649896243.png){kind=icon}
Abstract
We consider Bayesian inference for regression models of count data subject to underreporting. For the data generating process of counts as well as the fallible reporting process a joint model is specified, where the outcomes in both processes are related to a set of potential covariates. Identification of the joint model is achieved by additional information provided through validation data and incorporation of variable selection. For posterior inference we propose a convenient Markov chain Monte Carlo (MCMC) sampling scheme which relies on data augmentation and auxiliary mixture sampling techniques for this two-part model. Performance of the method is illustrated for simulated data and applied to analyse real data, collected to estimate risk of cervical cancer death.
References
| Amoros, E, Martin, J-L, Laumon, B (2006) Under-reporting of road crash casualties in France. Accident Analysis and Prevention, 38, 627–635. Google Scholar | Crossref | Medline | ISI | |
| Bratcher, TL, Stamey, JD (2002) Estimation of Poisson rates with misclassified counts. Biometrical Journal, 44, 946–956. Google Scholar | Crossref | ISI | |
| Cameron, AC, Trivedi, PK (2013) Regression Analysis of Count Data (2nd Edition). Cambridge: Cambridge University Press. Google Scholar | Crossref | |
| Dvorzak, M, Wagner, H (2015) pogit: Bayesian Variable Selection for a Poisson-Logistic Model. http://cran.r-project.org/ web/packages/pogit/index.html, R Package Version 1.0.0. Google Scholar | |
| Frühwirth-Schnatter, S, Frühwirth, R, Held, L, Rue, H (2009) Improved auxiliary mixture sampling for hierarchical models of non-Gaussian data. Statistics and Computing, 19, 479–492. Google Scholar | Crossref | ISI | |
| Fussl, A, Frühwirth-Schnatter, S, Frühwirth, R (2013) Efficient MCMC for binomial logit models. ACM Transactions on Modeling and Computer Simulation, 23, 1–21. Google Scholar | Crossref | ISI | |
| Gelman, A, Goegebeur, Y, Tuerlinckx, F, Van, Mechelen I (2000) Diagnostic checks for discrete data regression models using posterior predictive simulations. Applied Statistics, 49, 247–268. Google Scholar | ISI | |
| George, EI, McCulloch, RE (1993) Variable selection via Gibbs sampling. Journal of the American Statistical Association, 88, 881–889. Google Scholar | Crossref | ISI | |
| George, EI, McCulloch, RE (1997) Approaches for Bayesian variable selection. Statistica Sinica, 7, 339–373. Google Scholar | ISI | |
| Holmes, CC, Held, L (2006) Bayesian auxiliary variable models for binary and multinomial regression. Bayesian Analysis, 1, 145–168. Google Scholar | Crossref | ISI | |
| Ishwaran, H, Rao, SJ (2005) Spike and slab variable selection: Frequentist and Bayesian strategies. Annals of Statistics, 33, 730–773. Google Scholar | Crossref | ISI | |
| Li, T, Trivedi, PK, Guo, J (2003) Modeling response bias in count: A structural approach with an application to the national crime victimization survey data. Sociological Methods and Research, 31, 514–544. Google Scholar | SAGE Journals | ISI | |
| Little, RJA, Rubin, DB (2002) Statistical Analysis with Missing Data (2nd Edition). New York: John Wiley and Sons. Google Scholar | Crossref | |
| Liu, C (2004) Robit regression: A simple robust alternative to logistic and probit regression. In Gelman A and Meng X-L, eds. Applied Bayesian Modeling and Causal Inference from Incomplete-Data Perspectives, Chapter 21, pages 227–238. Chichester: John Wiley and Sons. Google Scholar | Crossref | |
| Ma, J, Li, Z (2010) Bayesian modeling of frequency-severity indeterminacy with an application to traffic crashes on two-lane highways. In Proceedings of the 10th International Conference of Chinese Transportation Professionals (ICCTP), Beijing, pages 1022–1033. Google Scholar | Crossref | |
| Mitchell, TJ, Beauchamp, JJ (1988) Bayesian variable selection in linear regression. Journal of the American Statistical Association, 83, 1023–1032. Google Scholar | Crossref | ISI | |
| Moreno, E, Girón, J (1998) Estimating with incomplete count data –A Bayesian approach. Journal of Statistical Planning and Inference, 66, 147–159. Google Scholar | Crossref | ISI | |
| Papadopoulos, G, Santos, Silva JMC (2012) Identification issues in some double-index models for non-negative data. Economics Letters, 117, 365–367. Google Scholar | Crossref | ISI | |
| Polson, NG, Scott, JG, Windle, J (2013) Bayesian inference for logistic models using Póly-Gamma latent variables. Journal of the American Statistical Association, 108, 1339–1349. Google Scholar | Crossref | ISI | |
| Powers, S, Gerlach, R, Stamey, J (2010) Bayesian variable selection for Poisson regression with underreported responses. Computational Statistics and Data Analysis, 54, 3289–3299. Google Scholar | Crossref | ISI | |
| Rubin, DB (1976) Inference and missing data. Biometrika, 63, 581–592. Google Scholar | Crossref | ISI | |
| Sposto, R, Preston, DL, Shimizu, Y, Mabuchi, K (1992) The effect of diagnostic misclassification on non-cancer and cancer mortality dose response in A-bomb survivors. Biometrics, 48, 605–617. Google Scholar | Crossref | Medline | ISI | |
| Stamey, JD, Young, DM, Seaman, JW Jr. (2008) A Bayesian approach to adjust for diagnostic misclassification between two mortality causes in Poisson regression. Statistics in Medicine, 27, 2440–2452. Google Scholar | Crossref | Medline | ISI | |
| Tüchler, R (2008) Bayesian variable selection for logistic models using auxiliary mixture sampling. Journal of Computational and Graphical Statistics, 17, 76–94. Google Scholar | Crossref | ISI | |
| Wagner, H, Duller, C (2012) Bayesian model selection for logistic regression models with random intercept. Computational Statistics and Data Analysis, 56, 1256–1274. Google Scholar | Crossref | ISI | |
| Whittemore, AS, Gong, G (1991) Poisson regression with misclassified counts: Application to cervical cancer mortality rates. Applied Statistics, 40, 81–93. Google Scholar | Crossref | ISI | |
| Winkelmann, R (1996) Markov Chain Monte Carlo analysis of underreported count data with an application to worker absenteeism. Empirical Economics, 21, 575–587. Google Scholar | Crossref | |
| Winkelmann, R, Zimmermann, KF (1993) Poisson-Logistic regression. Department of Economics, University of Munich, Working Paper No. 93–18. Google Scholar |
